LMFDB - Embedded newform 432.2.k.b.325.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,2,Mod(109,432)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("432.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	432.k (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.44953736732$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			325.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	432.325
Dual form			432.2.k.b.109.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +(0.707107 + 0.707107i) q^{5} -3.00000i q^{7} +2.82843i q^{8} +O(q^{10})$	\(q-1.41421i q^{2} -2.00000 q^{4} +(0.707107 + 0.707107i) q^{5} -3.00000i q^{7} +2.82843i q^{8} +(1.00000 - 1.00000i) q^{10} +(-3.53553 - 3.53553i) q^{11} +(1.00000 - 1.00000i) q^{13} -4.24264 q^{14} +4.00000 q^{16} -4.24264 q^{17} +(4.00000 - 4.00000i) q^{19} +(-1.41421 - 1.41421i) q^{20} +(-5.00000 + 5.00000i) q^{22} -2.82843i q^{23} -4.00000i q^{25} +(-1.41421 - 1.41421i) q^{26} +6.00000i q^{28} +(-2.82843 + 2.82843i) q^{29} -7.00000 q^{31} -5.65685i q^{32} +6.00000i q^{34} +(2.12132 - 2.12132i) q^{35} +(-2.00000 - 2.00000i) q^{37} +(-5.65685 - 5.65685i) q^{38} +(-2.00000 + 2.00000i) q^{40} +5.65685i q^{41} +(7.00000 + 7.00000i) q^{43} +(7.07107 + 7.07107i) q^{44} -4.00000 q^{46} +4.24264 q^{47} -2.00000 q^{49} -5.65685 q^{50} +(-2.00000 + 2.00000i) q^{52} +(0.707107 + 0.707107i) q^{53} -5.00000i q^{55} +8.48528 q^{56} +(4.00000 + 4.00000i) q^{58} +(7.07107 + 7.07107i) q^{59} +(10.0000 - 10.0000i) q^{61} +9.89949i q^{62} -8.00000 q^{64} +1.41421 q^{65} +(1.00000 - 1.00000i) q^{67} +8.48528 q^{68} +(-3.00000 - 3.00000i) q^{70} -15.5563i q^{71} +15.0000i q^{73} +(-2.82843 + 2.82843i) q^{74} +(-8.00000 + 8.00000i) q^{76} +(-10.6066 + 10.6066i) q^{77} +2.00000 q^{79} +(2.82843 + 2.82843i) q^{80} +8.00000 q^{82} +(7.77817 - 7.77817i) q^{83} +(-3.00000 - 3.00000i) q^{85} +(9.89949 - 9.89949i) q^{86} +(10.0000 - 10.0000i) q^{88} +1.41421i q^{89} +(-3.00000 - 3.00000i) q^{91} +5.65685i q^{92} -6.00000i q^{94} +5.65685 q^{95} -7.00000 q^{97} +2.82843i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} + 4 q^{10} + 4 q^{13} + 16 q^{16} + 16 q^{19} - 20 q^{22} - 28 q^{31} - 8 q^{37} - 8 q^{40} + 28 q^{43} - 16 q^{46} - 8 q^{49} - 8 q^{52} + 16 q^{58} + 40 q^{61} - 32 q^{64} + 4 q^{67} - 12 q^{70} - 32 q^{76} + 8 q^{79} + 32 q^{82} - 12 q^{85} + 40 q^{88} - 12 q^{91} - 28 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 4 * q^10 + 4 * q^13 + 16 * q^16 + 16 * q^19 - 20 * q^22 - 28 * q^31 - 8 * q^37 - 8 * q^40 + 28 * q^43 - 16 * q^46 - 8 * q^49 - 8 * q^52 + 16 * q^58 + 40 * q^61 - 32 * q^64 + 4 * q^67 - 12 * q^70 - 32 * q^76 + 8 * q^79 + 32 * q^82 - 12 * q^85 + 40 * q^88 - 12 * q^91 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.41421i		−	1.00000i
$2$		−	1.41421i		−	1.00000i
$3$		0			0
$3$		0			0
$4$	−2.00000			−1.00000
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i	0.847316	−	0.531089i	$-0.178217\pi$
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i	−0.531089	+	0.847316i	$0.678217\pi$
$6$		0			0
$7$		−	3.00000i		−	1.13389i	−0.823754	−	0.566947i	$-0.808125\pi$
$7$		−	3.00000i		−	1.13389i	0.823754	−	0.566947i	$-0.191875\pi$
$8$			2.82843i			1.00000i
$9$		0			0
$10$	1.00000	−	1.00000i	0.316228	−	0.316228i
$11$	−3.53553	−	3.53553i	−1.06600	−	1.06600i	−0.997662	−	0.0683416i	$-0.978229\pi$
$11$	−3.53553	−	3.53553i	−1.06600	−	1.06600i	−0.0683416	−	0.997662i	$-0.521771\pi$
$12$		0			0
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	0.832050	+	0.554700i	$0.187167\pi$
$14$	−4.24264			−1.13389
$15$		0			0
$16$	4.00000			1.00000
$17$	−4.24264			−1.02899			−0.514496	−	0.857493i	$-0.672021\pi$
$17$	−4.24264			−1.02899			−0.514496	+	0.857493i	$0.672021\pi$
$18$		0			0
$19$	4.00000	−	4.00000i	0.917663	−	0.917663i	−0.0791961	−	0.996859i	$-0.525235\pi$
$19$	4.00000	−	4.00000i	0.917663	−	0.917663i	0.996859	+	0.0791961i	$0.0252353\pi$
$20$	−1.41421	−	1.41421i	−0.316228	−	0.316228i
$21$		0			0
$22$	−5.00000	+	5.00000i	−1.06600	+	1.06600i
$23$		−	2.82843i		−	0.589768i	−0.955533	−	0.294884i	$-0.904719\pi$
$23$		−	2.82843i		−	0.589768i	0.955533	−	0.294884i	$-0.0952810\pi$
$24$		0			0
$25$		−	4.00000i		−	0.800000i
$26$	−1.41421	−	1.41421i	−0.277350	−	0.277350i
$27$		0			0
$28$			6.00000i			1.13389i
$29$	−2.82843	+	2.82843i	−0.525226	+	0.525226i	−0.919145	−	0.393919i	$-0.871119\pi$
$29$	−2.82843	+	2.82843i	−0.525226	+	0.525226i	0.393919	+	0.919145i	$0.371119\pi$
$30$		0			0
$31$	−7.00000			−1.25724			−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000			−1.25724			−0.628619	+	0.777714i	$0.716379\pi$
$32$		−	5.65685i		−	1.00000i
$33$		0			0
$34$			6.00000i			1.02899i
$35$	2.12132	−	2.12132i	0.358569	−	0.358569i
$36$		0			0
$37$	−2.00000	−	2.00000i	−0.328798	−	0.328798i	0.523331	−	0.852129i	$-0.324689\pi$
$37$	−2.00000	−	2.00000i	−0.328798	−	0.328798i	−0.852129	+	0.523331i	$0.824689\pi$
$38$	−5.65685	−	5.65685i	−0.917663	−	0.917663i
$39$		0			0
$40$	−2.00000	+	2.00000i	−0.316228	+	0.316228i
$41$			5.65685i			0.883452i	0.897150	+	0.441726i	$0.145634\pi$
$41$			5.65685i			0.883452i	−0.897150	+	0.441726i	$0.854366\pi$
$42$		0			0
$43$	7.00000	+	7.00000i	1.06749	+	1.06749i	0.997551	+	0.0699387i	$0.0222804\pi$
$43$	7.00000	+	7.00000i	1.06749	+	1.06749i	0.0699387	+	0.997551i	$0.477720\pi$
$44$	7.07107	+	7.07107i	1.06600	+	1.06600i
$45$		0			0
$46$	−4.00000			−0.589768
$47$	4.24264			0.618853			0.309426	−	0.950923i	$-0.399863\pi$
$47$	4.24264			0.618853			0.309426	+	0.950923i	$0.399863\pi$
$48$		0			0
$49$	−2.00000			−0.285714
$50$	−5.65685			−0.800000
$51$		0			0
$52$	−2.00000	+	2.00000i	−0.277350	+	0.277350i
$53$	0.707107	+	0.707107i	0.0971286	+	0.0971286i	0.754001	−	0.656873i	$-0.228121\pi$
$53$	0.707107	+	0.707107i	0.0971286	+	0.0971286i	−0.656873	+	0.754001i	$0.728121\pi$
$54$		0			0
$55$		−	5.00000i		−	0.674200i
$56$	8.48528			1.13389
$57$		0			0
$58$	4.00000	+	4.00000i	0.525226	+	0.525226i
$59$	7.07107	+	7.07107i	0.920575	+	0.920575i	0.997070	−	0.0764953i	$-0.0243730\pi$
$59$	7.07107	+	7.07107i	0.920575	+	0.920575i	−0.0764953	+	0.997070i	$0.524373\pi$
$60$		0			0
$61$	10.0000	−	10.0000i	1.28037	−	1.28037i	0.339911	−	0.940457i	$-0.389603\pi$
$61$	10.0000	−	10.0000i	1.28037	−	1.28037i	0.940457	−	0.339911i	$-0.110397\pi$
$62$			9.89949i			1.25724i
$63$		0			0
$64$	−8.00000			−1.00000
$65$	1.41421			0.175412
$66$		0			0
$67$	1.00000	−	1.00000i	0.122169	−	0.122169i	−0.643379	−	0.765548i	$-0.722468\pi$
$67$	1.00000	−	1.00000i	0.122169	−	0.122169i	0.765548	+	0.643379i	$0.222468\pi$
$68$	8.48528			1.02899
$69$		0			0
$70$	−3.00000	−	3.00000i	−0.358569	−	0.358569i
$71$		−	15.5563i		−	1.84620i	−0.384561	−	0.923099i	$-0.625647\pi$
$71$		−	15.5563i		−	1.84620i	0.384561	−	0.923099i	$-0.374353\pi$
$72$		0			0
$73$			15.0000i			1.75562i	0.479012	+	0.877809i	$0.340995\pi$
$73$			15.0000i			1.75562i	−0.479012	+	0.877809i	$0.659005\pi$
$74$	−2.82843	+	2.82843i	−0.328798	+	0.328798i
$75$		0			0
$76$	−8.00000	+	8.00000i	−0.917663	+	0.917663i
$77$	−10.6066	+	10.6066i	−1.20873	+	1.20873i
$78$		0			0
$79$	2.00000			0.225018			0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000			0.225018			0.112509	+	0.993651i	$0.464111\pi$
$80$	2.82843	+	2.82843i	0.316228	+	0.316228i
$81$		0			0
$82$	8.00000			0.883452
$83$	7.77817	−	7.77817i	0.853766	−	0.853766i	−0.136829	−	0.990595i	$-0.543691\pi$
$83$	7.77817	−	7.77817i	0.853766	−	0.853766i	0.990595	+	0.136829i	$0.0436911\pi$
$84$		0			0
$85$	−3.00000	−	3.00000i	−0.325396	−	0.325396i
$86$	9.89949	−	9.89949i	1.06749	−	1.06749i
$87$		0			0
$88$	10.0000	−	10.0000i	1.06600	−	1.06600i
$89$			1.41421i			0.149906i	0.997187	+	0.0749532i	$0.0238807\pi$
$89$			1.41421i			0.149906i	−0.997187	+	0.0749532i	$0.976119\pi$
$90$		0			0
$91$	−3.00000	−	3.00000i	−0.314485	−	0.314485i
$92$			5.65685i			0.589768i
$93$		0			0
$94$		−	6.00000i		−	0.618853i
$95$	5.65685			0.580381
$96$		0			0
$97$	−7.00000			−0.710742			−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000			−0.710742			−0.355371	+	0.934725i	$0.615646\pi$
$98$			2.82843i			0.285714i
$99$		0			0
$100$			8.00000i			0.800000i
$101$	4.94975	+	4.94975i	0.492518	+	0.492518i	0.909099	−	0.416581i	$-0.136772\pi$
$101$	4.94975	+	4.94975i	0.492518	+	0.492518i	−0.416581	+	0.909099i	$0.636772\pi$
$102$		0			0
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$	2.82843	+	2.82843i	0.277350	+	0.277350i
$105$		0			0
$106$	1.00000	−	1.00000i	0.0971286	−	0.0971286i
$107$	−7.77817	−	7.77817i	−0.751945	−	0.751945i	0.222897	−	0.974842i	$-0.428449\pi$
$107$	−7.77817	−	7.77817i	−0.751945	−	0.751945i	−0.974842	+	0.222897i	$0.928449\pi$
$108$		0			0
$109$	10.0000	−	10.0000i	0.957826	−	0.957826i	−0.0413197	−	0.999146i	$-0.513156\pi$
$109$	10.0000	−	10.0000i	0.957826	−	0.957826i	0.999146	+	0.0413197i	$0.0131562\pi$
$110$	−7.07107			−0.674200
$111$		0			0
$112$		−	12.0000i		−	1.13389i
$113$	8.48528			0.798228			0.399114	−	0.916901i	$-0.369318\pi$
$113$	8.48528			0.798228			0.399114	+	0.916901i	$0.369318\pi$
$114$		0			0
$115$	2.00000	−	2.00000i	0.186501	−	0.186501i
$116$	5.65685	−	5.65685i	0.525226	−	0.525226i
$117$		0			0
$118$	10.0000	−	10.0000i	0.920575	−	0.920575i
$119$			12.7279i			1.16677i
$120$		0			0
$121$			14.0000i			1.27273i
$122$	−14.1421	−	14.1421i	−1.28037	−	1.28037i
$123$		0			0
$124$	14.0000			1.25724
$125$	6.36396	−	6.36396i	0.569210	−	0.569210i
$126$		0			0
$127$	−13.0000			−1.15356			−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000			−1.15356			−0.576782	+	0.816898i	$0.695692\pi$
$128$			11.3137i			1.00000i
$129$		0			0
$130$		−	2.00000i		−	0.175412i
$131$	−0.707107	+	0.707107i	−0.0617802	+	0.0617802i	−0.737322	−	0.675542i	$-0.763910\pi$
$131$	−0.707107	+	0.707107i	−0.0617802	+	0.0617802i	0.675542	+	0.737322i	$0.263910\pi$
$132$		0			0
$133$	−12.0000	−	12.0000i	−1.04053	−	1.04053i
$134$	−1.41421	−	1.41421i	−0.122169	−	0.122169i
$135$		0			0
$136$		−	12.0000i		−	1.02899i
$137$			18.3848i			1.57072i	0.619041	+	0.785359i	$0.287521\pi$
$137$			18.3848i			1.57072i	−0.619041	+	0.785359i	$0.712479\pi$
$138$		0			0
$139$	10.0000	+	10.0000i	0.848189	+	0.848189i	0.989907	−	0.141718i	$-0.0452627\pi$
$139$	10.0000	+	10.0000i	0.848189	+	0.848189i	−0.141718	+	0.989907i	$0.545263\pi$
$140$	−4.24264	+	4.24264i	−0.358569	+	0.358569i
$141$		0			0
$142$	−22.0000			−1.84620
$143$	−7.07107			−0.591312
$144$		0			0
$145$	−4.00000			−0.332182
$146$	21.2132			1.75562
$147$		0			0
$148$	4.00000	+	4.00000i	0.328798	+	0.328798i
$149$	4.94975	+	4.94975i	0.405499	+	0.405499i	0.880166	−	0.474667i	$-0.157431\pi$
$149$	4.94975	+	4.94975i	0.405499	+	0.405499i	−0.474667	+	0.880166i	$0.657431\pi$
$150$		0			0
$151$			21.0000i			1.70896i	0.519488	+	0.854478i	$0.326123\pi$
$151$			21.0000i			1.70896i	−0.519488	+	0.854478i	$0.673877\pi$
$152$	11.3137	+	11.3137i	0.917663	+	0.917663i
$153$		0			0
$154$	15.0000	+	15.0000i	1.20873	+	1.20873i
$155$	−4.94975	−	4.94975i	−0.397573	−	0.397573i
$156$		0			0
$157$	−2.00000	+	2.00000i	−0.159617	+	0.159617i	−0.782397	−	0.622780i	$-0.786003\pi$
$157$	−2.00000	+	2.00000i	−0.159617	+	0.159617i	0.622780	+	0.782397i	$0.286003\pi$
$158$		−	2.82843i		−	0.225018i
$159$		0			0
$160$	4.00000	−	4.00000i	0.316228	−	0.316228i
$161$	−8.48528			−0.668734
$162$		0			0
$163$	−5.00000	+	5.00000i	−0.391630	+	0.391630i	−0.875268	−	0.483638i	$-0.839315\pi$
$163$	−5.00000	+	5.00000i	−0.391630	+	0.391630i	0.483638	+	0.875268i	$0.339315\pi$
$164$		−	11.3137i		−	0.883452i
$165$		0			0
$166$	−11.0000	−	11.0000i	−0.853766	−	0.853766i
$167$		−	7.07107i		−	0.547176i	−0.961847	−	0.273588i	$-0.911790\pi$
$167$		−	7.07107i		−	0.547176i	0.961847	−	0.273588i	$-0.0882104\pi$
$168$		0			0
$169$			11.0000i			0.846154i
$170$	−4.24264	+	4.24264i	−0.325396	+	0.325396i
$171$		0			0
$172$	−14.0000	−	14.0000i	−1.06749	−	1.06749i
$173$	7.77817	−	7.77817i	0.591364	−	0.591364i	−0.346636	−	0.938000i	$-0.612676\pi$
$173$	7.77817	−	7.77817i	0.591364	−	0.591364i	0.938000	+	0.346636i	$0.112676\pi$
$174$		0			0
$175$	−12.0000			−0.907115
$176$	−14.1421	−	14.1421i	−1.06600	−	1.06600i
$177$		0			0
$178$	2.00000			0.149906
$179$	7.77817	−	7.77817i	0.581368	−	0.581368i	−0.353911	−	0.935279i	$-0.615148\pi$
$179$	7.77817	−	7.77817i	0.581368	−	0.581368i	0.935279	+	0.353911i	$0.115148\pi$
$180$		0			0
$181$	16.0000	+	16.0000i	1.18927	+	1.18927i	0.977269	+	0.212001i	$0.0679981\pi$
$181$	16.0000	+	16.0000i	1.18927	+	1.18927i	0.212001	+	0.977269i	$0.432002\pi$
$182$	−4.24264	+	4.24264i	−0.314485	+	0.314485i
$183$		0			0
$184$	8.00000			0.589768
$185$		−	2.82843i		−	0.207950i
$186$		0			0
$187$	15.0000	+	15.0000i	1.09691	+	1.09691i
$188$	−8.48528			−0.618853
$189$		0			0
$190$		−	8.00000i		−	0.580381i
$191$	4.24264			0.306987			0.153493	−	0.988150i	$-0.450948\pi$
$191$	4.24264			0.306987			0.153493	+	0.988150i	$0.450948\pi$
$192$		0			0
$193$	−7.00000			−0.503871			−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000			−0.503871			−0.251936	+	0.967744i	$0.581067\pi$
$194$			9.89949i			0.710742i
$195$		0			0
$196$	4.00000			0.285714
$197$	−16.2635	−	16.2635i	−1.15872	−	1.15872i	−0.984750	−	0.173973i	$-0.944340\pi$
$197$	−16.2635	−	16.2635i	−1.15872	−	1.15872i	−0.173973	−	0.984750i	$-0.555660\pi$
$198$		0			0
$199$		−	15.0000i		−	1.06332i	−0.846957	−	0.531661i	$-0.821568\pi$
$199$		−	15.0000i		−	1.06332i	0.846957	−	0.531661i	$-0.178432\pi$
$200$	11.3137			0.800000
$201$		0			0
$202$	7.00000	−	7.00000i	0.492518	−	0.492518i
$203$	8.48528	+	8.48528i	0.595550	+	0.595550i
$204$		0			0
$205$	−4.00000	+	4.00000i	−0.279372	+	0.279372i
$206$	−8.48528			−0.591198
$207$		0			0
$208$	4.00000	−	4.00000i	0.277350	−	0.277350i
$209$	−28.2843			−1.95646
$210$		0			0
$211$	4.00000	−	4.00000i	0.275371	−	0.275371i	−0.555887	−	0.831258i	$-0.687621\pi$
$211$	4.00000	−	4.00000i	0.275371	−	0.275371i	0.831258	+	0.555887i	$0.187621\pi$
$212$	−1.41421	−	1.41421i	−0.0971286	−	0.0971286i
$213$		0			0
$214$	−11.0000	+	11.0000i	−0.751945	+	0.751945i
$215$			9.89949i			0.675140i
$216$		0			0
$217$			21.0000i			1.42557i
$218$	−14.1421	−	14.1421i	−0.957826	−	0.957826i
$219$		0			0
$220$			10.0000i			0.674200i
$221$	−4.24264	+	4.24264i	−0.285391	+	0.285391i
$222$		0			0
$223$	−22.0000			−1.47323			−0.736614	−	0.676313i	$-0.763577\pi$
$223$	−22.0000			−1.47323			−0.736614	+	0.676313i	$0.763577\pi$
$224$	−16.9706			−1.13389
$225$		0			0
$226$		−	12.0000i		−	0.798228i
$227$	−7.07107	+	7.07107i	−0.469323	+	0.469323i	−0.901695	−	0.432372i	$-0.857677\pi$
$227$	−7.07107	+	7.07107i	−0.469323	+	0.469323i	0.432372	+	0.901695i	$0.357677\pi$
$228$		0			0
$229$	7.00000	+	7.00000i	0.462573	+	0.462573i	0.899498	−	0.436925i	$-0.143932\pi$
$229$	7.00000	+	7.00000i	0.462573	+	0.462573i	−0.436925	+	0.899498i	$0.643932\pi$
$230$	−2.82843	−	2.82843i	−0.186501	−	0.186501i
$231$		0			0
$232$	−8.00000	−	8.00000i	−0.525226	−	0.525226i
$233$		−	15.5563i		−	1.01913i	−0.860432	−	0.509565i	$-0.829806\pi$
$233$		−	15.5563i		−	1.01913i	0.860432	−	0.509565i	$-0.170194\pi$
$234$		0			0
$235$	3.00000	+	3.00000i	0.195698	+	0.195698i
$236$	−14.1421	−	14.1421i	−0.920575	−	0.920575i
$237$		0			0
$238$	18.0000			1.16677
$239$	−25.4558			−1.64660			−0.823301	−	0.567605i	$-0.807870\pi$
$239$	−25.4558			−1.64660			−0.823301	+	0.567605i	$0.807870\pi$
$240$		0			0
$241$	−4.00000			−0.257663			−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000			−0.257663			−0.128831	+	0.991667i	$0.541123\pi$
$242$	19.7990			1.27273
$243$		0			0
$244$	−20.0000	+	20.0000i	−1.28037	+	1.28037i
$245$	−1.41421	−	1.41421i	−0.0903508	−	0.0903508i
$246$		0			0
$247$		−	8.00000i		−	0.509028i
$248$		−	19.7990i		−	1.25724i
$249$		0			0
$250$	−9.00000	−	9.00000i	−0.569210	−	0.569210i
$251$	−1.41421	−	1.41421i	−0.0892644	−	0.0892644i	0.661065	−	0.750329i	$-0.270105\pi$
$251$	−1.41421	−	1.41421i	−0.0892644	−	0.0892644i	−0.750329	+	0.661065i	$0.770105\pi$
$252$		0			0
$253$	−10.0000	+	10.0000i	−0.628695	+	0.628695i
$254$			18.3848i			1.15356i
$255$		0			0
$256$	16.0000			1.00000
$257$	29.6985			1.85254			0.926270	−	0.376860i	$-0.122996\pi$
$257$	29.6985			1.85254			0.926270	+	0.376860i	$0.122996\pi$
$258$		0			0
$259$	−6.00000	+	6.00000i	−0.372822	+	0.372822i
$260$	−2.82843			−0.175412
$261$		0			0
$262$	1.00000	+	1.00000i	0.0617802	+	0.0617802i
$263$			5.65685i			0.348817i	0.984673	+	0.174408i	$0.0558013\pi$
$263$			5.65685i			0.348817i	−0.984673	+	0.174408i	$0.944199\pi$
$264$		0			0
$265$			1.00000i			0.0614295i
$266$	−16.9706	+	16.9706i	−1.04053	+	1.04053i
$267$		0			0
$268$	−2.00000	+	2.00000i	−0.122169	+	0.122169i
$269$	−11.3137	+	11.3137i	−0.689809	+	0.689809i	−0.962190	−	0.272381i	$-0.912189\pi$
$269$	−11.3137	+	11.3137i	−0.689809	+	0.689809i	0.272381	+	0.962190i	$0.412189\pi$
$270$		0			0
$271$	−13.0000			−0.789694			−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000			−0.789694			−0.394847	+	0.918747i	$0.629202\pi$
$272$	−16.9706			−1.02899
$273$		0			0
$274$	26.0000			1.57072
$275$	−14.1421	+	14.1421i	−0.852803	+	0.852803i
$276$		0			0
$277$	−20.0000	−	20.0000i	−1.20168	−	1.20168i	−0.973654	−	0.228029i	$-0.926772\pi$
$277$	−20.0000	−	20.0000i	−1.20168	−	1.20168i	−0.228029	−	0.973654i	$-0.573228\pi$
$278$	14.1421	−	14.1421i	0.848189	−	0.848189i
$279$		0			0
$280$	6.00000	+	6.00000i	0.358569	+	0.358569i
$281$		−	24.0416i		−	1.43420i	−0.696969	−	0.717102i	$-0.745468\pi$
$281$		−	24.0416i		−	1.43420i	0.696969	−	0.717102i	$-0.254532\pi$
$282$		0			0
$283$	4.00000	+	4.00000i	0.237775	+	0.237775i	0.815928	−	0.578153i	$-0.196226\pi$
$283$	4.00000	+	4.00000i	0.237775	+	0.237775i	−0.578153	+	0.815928i	$0.696226\pi$
$284$			31.1127i			1.84620i
$285$		0			0
$286$			10.0000i			0.591312i
$287$	16.9706			1.00174
$288$		0			0
$289$	1.00000			0.0588235
$290$			5.65685i			0.332182i
$291$		0			0
$292$		−	30.0000i		−	1.75562i
$293$	−14.1421	−	14.1421i	−0.826192	−	0.826192i	0.160795	−	0.986988i	$-0.448594\pi$
$293$	−14.1421	−	14.1421i	−0.826192	−	0.826192i	−0.986988	+	0.160795i	$0.948594\pi$
$294$		0			0
$295$			10.0000i			0.582223i
$296$	5.65685	−	5.65685i	0.328798	−	0.328798i
$297$		0			0
$298$	7.00000	−	7.00000i	0.405499	−	0.405499i
$299$	−2.82843	−	2.82843i	−0.163572	−	0.163572i
$300$		0			0
$301$	21.0000	−	21.0000i	1.21042	−	1.21042i
$302$	29.6985			1.70896
$303$		0			0
$304$	16.0000	−	16.0000i	0.917663	−	0.917663i
$305$	14.1421			0.809776
$306$		0			0
$307$	7.00000	−	7.00000i	0.399511	−	0.399511i	−0.478549	−	0.878061i	$-0.658837\pi$
$307$	7.00000	−	7.00000i	0.399511	−	0.399511i	0.878061	+	0.478549i	$0.158837\pi$
$308$	21.2132	−	21.2132i	1.20873	−	1.20873i
$309$		0			0
$310$	−7.00000	+	7.00000i	−0.397573	+	0.397573i
$311$		−	19.7990i		−	1.12270i	−0.827579	−	0.561349i	$-0.810283\pi$
$311$		−	19.7990i		−	1.12270i	0.827579	−	0.561349i	$-0.189717\pi$
$312$		0			0
$313$		−	27.0000i		−	1.52613i	−0.646322	−	0.763065i	$-0.723694\pi$
$313$		−	27.0000i		−	1.52613i	0.646322	−	0.763065i	$-0.276306\pi$
$314$	2.82843	+	2.82843i	0.159617	+	0.159617i
$315$		0			0
$316$	−4.00000			−0.225018
$317$	24.7487	−	24.7487i	1.39003	−	1.39003i	0.564799	−	0.825228i	$-0.308954\pi$
$317$	24.7487	−	24.7487i	1.39003	−	1.39003i	0.825228	−	0.564799i	$-0.191046\pi$
$318$		0			0
$319$	20.0000			1.11979
$320$	−5.65685	−	5.65685i	−0.316228	−	0.316228i
$321$		0			0
$322$			12.0000i			0.668734i
$323$	−16.9706	+	16.9706i	−0.944267	+	0.944267i
$324$		0			0
$325$	−4.00000	−	4.00000i	−0.221880	−	0.221880i
$326$	7.07107	+	7.07107i	0.391630	+	0.391630i
$327$		0			0
$328$	−16.0000			−0.883452
$329$		−	12.7279i		−	0.701713i
$330$		0			0
$331$	7.00000	+	7.00000i	0.384755	+	0.384755i	0.872812	−	0.488057i	$-0.162294\pi$
$331$	7.00000	+	7.00000i	0.384755	+	0.384755i	−0.488057	+	0.872812i	$0.662294\pi$
$332$	−15.5563	+	15.5563i	−0.853766	+	0.853766i
$333$		0			0
$334$	−10.0000			−0.547176
$335$	1.41421			0.0772667
$336$		0			0
$337$	20.0000			1.08947			0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000			1.08947			0.544735	+	0.838608i	$0.316630\pi$
$338$	15.5563			0.846154
$339$		0			0
$340$	6.00000	+	6.00000i	0.325396	+	0.325396i
$341$	24.7487	+	24.7487i	1.34022	+	1.34022i
$342$		0			0
$343$		−	15.0000i		−	0.809924i
$344$	−19.7990	+	19.7990i	−1.06749	+	1.06749i
$345$		0			0
$346$	−11.0000	−	11.0000i	−0.591364	−	0.591364i
$347$	17.6777	+	17.6777i	0.948987	+	0.948987i	0.998761	−	0.0497735i	$-0.0158499\pi$
$347$	17.6777	+	17.6777i	0.948987	+	0.948987i	−0.0497735	+	0.998761i	$0.515850\pi$
$348$		0			0
$349$	4.00000	−	4.00000i	0.214115	−	0.214115i	−0.591898	−	0.806013i	$-0.701621\pi$
$349$	4.00000	−	4.00000i	0.214115	−	0.214115i	0.806013	+	0.591898i	$0.201621\pi$
$350$			16.9706i			0.907115i
$351$		0			0
$352$	−20.0000	+	20.0000i	−1.06600	+	1.06600i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	11.0000	−	11.0000i	0.583819	−	0.583819i
$356$		−	2.82843i		−	0.149906i
$357$		0			0
$358$	−11.0000	−	11.0000i	−0.581368	−	0.581368i
$359$			35.3553i			1.86598i	0.359898	+	0.932992i	$0.382812\pi$
$359$			35.3553i			1.86598i	−0.359898	+	0.932992i	$0.617188\pi$
$360$		0			0
$361$		−	13.0000i		−	0.684211i
$362$	22.6274	−	22.6274i	1.18927	−	1.18927i
$363$		0			0
$364$	6.00000	+	6.00000i	0.314485	+	0.314485i
$365$	−10.6066	+	10.6066i	−0.555175	+	0.555175i
$366$		0			0
$367$	11.0000			0.574195			0.287098	−	0.957901i	$-0.407310\pi$
$367$	11.0000			0.574195			0.287098	+	0.957901i	$0.407310\pi$
$368$		−	11.3137i		−	0.589768i
$369$		0			0
$370$	−4.00000			−0.207950
$371$	2.12132	−	2.12132i	0.110133	−	0.110133i
$372$		0			0
$373$	1.00000	+	1.00000i	0.0517780	+	0.0517780i	0.732522	−	0.680744i	$-0.238343\pi$
$373$	1.00000	+	1.00000i	0.0517780	+	0.0517780i	−0.680744	+	0.732522i	$0.738343\pi$
$374$	21.2132	−	21.2132i	1.09691	−	1.09691i
$375$		0			0
$376$			12.0000i			0.618853i
$377$			5.65685i			0.291343i
$378$		0			0
$379$	−2.00000	−	2.00000i	−0.102733	−	0.102733i	0.653872	−	0.756605i	$-0.273143\pi$
$379$	−2.00000	−	2.00000i	−0.102733	−	0.102733i	−0.756605	+	0.653872i	$0.773143\pi$
$380$	−11.3137			−0.580381
$381$		0			0
$382$		−	6.00000i		−	0.306987i
$383$	−4.24264			−0.216789			−0.108394	−	0.994108i	$-0.534571\pi$
$383$	−4.24264			−0.216789			−0.108394	+	0.994108i	$0.534571\pi$
$384$		0			0
$385$	−15.0000			−0.764471
$386$			9.89949i			0.503871i
$387$		0			0
$388$	14.0000			0.710742
$389$	9.19239	+	9.19239i	0.466073	+	0.466073i	0.900640	−	0.434567i	$-0.143099\pi$
$389$	9.19239	+	9.19239i	0.466073	+	0.466073i	−0.434567	+	0.900640i	$0.643099\pi$
$390$		0			0
$391$			12.0000i			0.606866i
$392$		−	5.65685i		−	0.285714i
$393$		0			0
$394$	−23.0000	+	23.0000i	−1.15872	+	1.15872i
$395$	1.41421	+	1.41421i	0.0711568	+	0.0711568i
$396$		0			0
$397$	−5.00000	+	5.00000i	−0.250943	+	0.250943i	−0.821357	−	0.570414i	$-0.806783\pi$
$397$	−5.00000	+	5.00000i	−0.250943	+	0.250943i	0.570414	+	0.821357i	$0.306783\pi$
$398$	−21.2132			−1.06332
$399$		0			0
$400$		−	16.0000i		−	0.800000i
$401$	8.48528			0.423735			0.211867	−	0.977298i	$-0.432046\pi$
$401$	8.48528			0.423735			0.211867	+	0.977298i	$0.432046\pi$
$402$		0			0
$403$	−7.00000	+	7.00000i	−0.348695	+	0.348695i
$404$	−9.89949	−	9.89949i	−0.492518	−	0.492518i
$405$		0			0
$406$	12.0000	−	12.0000i	0.595550	−	0.595550i
$407$			14.1421i			0.701000i
$408$		0			0
$409$		−	3.00000i		−	0.148340i	−0.997246	−	0.0741702i	$-0.976369\pi$
$409$		−	3.00000i		−	0.148340i	0.997246	−	0.0741702i	$-0.0236308\pi$
$410$	5.65685	+	5.65685i	0.279372	+	0.279372i
$411$		0			0
$412$			12.0000i			0.591198i
$413$	21.2132	−	21.2132i	1.04383	−	1.04383i
$414$		0			0
$415$	11.0000			0.539969
$416$	−5.65685	−	5.65685i	−0.277350	−	0.277350i
$417$		0			0
$418$			40.0000i			1.95646i
$419$	−7.07107	+	7.07107i	−0.345444	+	0.345444i	−0.858409	−	0.512965i	$-0.828547\pi$
$419$	−7.07107	+	7.07107i	−0.345444	+	0.345444i	0.512965	+	0.858409i	$0.328547\pi$
$420$		0			0
$421$	16.0000	+	16.0000i	0.779792	+	0.779792i	0.979795	−	0.200003i	$-0.0640952\pi$
$421$	16.0000	+	16.0000i	0.779792	+	0.779792i	−0.200003	+	0.979795i	$0.564095\pi$
$422$	−5.65685	−	5.65685i	−0.275371	−	0.275371i
$423$		0			0
$424$	−2.00000	+	2.00000i	−0.0971286	+	0.0971286i
$425$			16.9706i			0.823193i
$426$		0			0
$427$	−30.0000	−	30.0000i	−1.45180	−	1.45180i
$428$	15.5563	+	15.5563i	0.751945	+	0.751945i
$429$		0			0
$430$	14.0000			0.675140
$431$	−4.24264			−0.204361			−0.102180	−	0.994766i	$-0.532582\pi$
$431$	−4.24264			−0.204361			−0.102180	+	0.994766i	$0.532582\pi$
$432$		0			0
$433$	5.00000			0.240285			0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000			0.240285			0.120142	+	0.992757i	$0.461665\pi$
$434$	29.6985			1.42557
$435$		0			0
$436$	−20.0000	+	20.0000i	−0.957826	+	0.957826i
$437$	−11.3137	−	11.3137i	−0.541208	−	0.541208i
$438$		0			0
$439$		−	15.0000i		−	0.715911i	−0.933739	−	0.357955i	$-0.883474\pi$
$439$		−	15.0000i		−	0.715911i	0.933739	−	0.357955i	$-0.116526\pi$
$440$	14.1421			0.674200
$441$		0			0
$442$	6.00000	+	6.00000i	0.285391	+	0.285391i
$443$	−9.89949	−	9.89949i	−0.470339	−	0.470339i	0.431685	−	0.902024i	$-0.357919\pi$
$443$	−9.89949	−	9.89949i	−0.470339	−	0.470339i	−0.902024	+	0.431685i	$0.857919\pi$
$444$		0			0
$445$	−1.00000	+	1.00000i	−0.0474045	+	0.0474045i
$446$			31.1127i			1.47323i
$447$		0			0
$448$			24.0000i			1.13389i
$449$	16.9706			0.800890			0.400445	−	0.916321i	$-0.368855\pi$
$449$	16.9706			0.800890			0.400445	+	0.916321i	$0.368855\pi$
$450$		0			0
$451$	20.0000	−	20.0000i	0.941763	−	0.941763i
$452$	−16.9706			−0.798228
$453$		0			0
$454$	10.0000	+	10.0000i	0.469323	+	0.469323i
$455$		−	4.24264i		−	0.198898i
$456$		0			0
$457$			3.00000i			0.140334i	0.997535	+	0.0701670i	$0.0223532\pi$
$457$			3.00000i			0.140334i	−0.997535	+	0.0701670i	$0.977647\pi$
$458$	9.89949	−	9.89949i	0.462573	−	0.462573i
$459$		0			0
$460$	−4.00000	+	4.00000i	−0.186501	+	0.186501i
$461$	24.7487	−	24.7487i	1.15266	−	1.15266i	0.166647	−	0.986017i	$-0.446706\pi$
$461$	24.7487	−	24.7487i	1.15266	−	1.15266i	0.986017	−	0.166647i	$-0.0532940\pi$
$462$		0			0
$463$	−7.00000			−0.325318			−0.162659	−	0.986682i	$-0.552007\pi$
$463$	−7.00000			−0.325318			−0.162659	+	0.986682i	$0.552007\pi$
$464$	−11.3137	+	11.3137i	−0.525226	+	0.525226i
$465$		0			0
$466$	−22.0000			−1.01913
$467$	3.53553	−	3.53553i	0.163605	−	0.163605i	−0.620557	−	0.784162i	$-0.713093\pi$
$467$	3.53553	−	3.53553i	0.163605	−	0.163605i	0.784162	+	0.620557i	$0.213093\pi$
$468$		0			0
$469$	−3.00000	−	3.00000i	−0.138527	−	0.138527i
$470$	4.24264	−	4.24264i	0.195698	−	0.195698i
$471$		0			0
$472$	−20.0000	+	20.0000i	−0.920575	+	0.920575i
$473$		−	49.4975i		−	2.27590i
$474$		0			0
$475$	−16.0000	−	16.0000i	−0.734130	−	0.734130i
$476$		−	25.4558i		−	1.16677i
$477$		0			0
$478$			36.0000i			1.64660i
$479$	−4.24264			−0.193851			−0.0969256	−	0.995292i	$-0.530901\pi$
$479$	−4.24264			−0.193851			−0.0969256	+	0.995292i	$0.530901\pi$
$480$		0			0
$481$	−4.00000			−0.182384
$482$			5.65685i			0.257663i
$483$		0			0
$484$		−	28.0000i		−	1.27273i
$485$	−4.94975	−	4.94975i	−0.224756	−	0.224756i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	28.2843	+	28.2843i	1.28037	+	1.28037i
$489$		0			0
$490$	−2.00000	+	2.00000i	−0.0903508	+	0.0903508i
$491$	4.94975	+	4.94975i	0.223379	+	0.223379i	0.809920	−	0.586541i	$-0.199511\pi$
$491$	4.94975	+	4.94975i	0.223379	+	0.223379i	−0.586541	+	0.809920i	$0.699511\pi$
$492$		0			0
$493$	12.0000	−	12.0000i	0.540453	−	0.540453i
$494$	−11.3137			−0.509028
$495$		0			0
$496$	−28.0000			−1.25724
$497$	−46.6690			−2.09339
$498$		0			0
$499$	−23.0000	+	23.0000i	−1.02962	+	1.02962i	−0.0300737	+	0.999548i	$0.509574\pi$
$499$	−23.0000	+	23.0000i	−1.02962	+	1.02962i	−0.999548	+	0.0300737i	$0.990426\pi$
$500$	−12.7279	+	12.7279i	−0.569210	+	0.569210i
$501$		0			0
$502$	−2.00000	+	2.00000i	−0.0892644	+	0.0892644i
$503$			22.6274i			1.00891i	0.863439	+	0.504453i	$0.168306\pi$
$503$			22.6274i			1.00891i	−0.863439	+	0.504453i	$0.831694\pi$
$504$		0			0
$505$			7.00000i			0.311496i
$506$	14.1421	+	14.1421i	0.628695	+	0.628695i
$507$		0			0
$508$	26.0000			1.15356
$509$	−4.94975	+	4.94975i	−0.219394	+	0.219394i	−0.808243	−	0.588849i	$-0.799581\pi$
$509$	−4.94975	+	4.94975i	−0.219394	+	0.219394i	0.588849	+	0.808243i	$0.299581\pi$
$510$		0			0
$511$	45.0000			1.99068
$512$		−	22.6274i		−	1.00000i
$513$		0			0
$514$		−	42.0000i		−	1.85254i
$515$	4.24264	−	4.24264i	0.186953	−	0.186953i
$516$		0			0
$517$	−15.0000	−	15.0000i	−0.659699	−	0.659699i
$518$	8.48528	+	8.48528i	0.372822	+	0.372822i
$519$		0			0
$520$			4.00000i			0.175412i
$521$			26.8701i			1.17720i	0.808425	+	0.588599i	$0.200320\pi$
$521$			26.8701i			1.17720i	−0.808425	+	0.588599i	$0.799680\pi$
$522$		0			0
$523$	−23.0000	−	23.0000i	−1.00572	−	1.00572i	−0.999984	−	0.00573623i	$-0.998174\pi$
$523$	−23.0000	−	23.0000i	−1.00572	−	1.00572i	−0.00573623	−	0.999984i	$-0.501826\pi$
$524$	1.41421	−	1.41421i	0.0617802	−	0.0617802i
$525$		0			0
$526$	8.00000			0.348817
$527$	29.6985			1.29369
$528$		0			0
$529$	15.0000			0.652174
$530$	1.41421			0.0614295
$531$		0			0
$532$	24.0000	+	24.0000i	1.04053	+	1.04053i
$533$	5.65685	+	5.65685i	0.245026	+	0.245026i
$534$		0			0
$535$		−	11.0000i		−	0.475571i
$536$	2.82843	+	2.82843i	0.122169	+	0.122169i
$537$		0			0
$538$	16.0000	+	16.0000i	0.689809	+	0.689809i
$539$	7.07107	+	7.07107i	0.304572	+	0.304572i
$540$		0			0
$541$	−2.00000	+	2.00000i	−0.0859867	+	0.0859867i	−0.748792	−	0.662805i	$-0.769366\pi$
$541$	−2.00000	+	2.00000i	−0.0859867	+	0.0859867i	0.662805	+	0.748792i	$0.269366\pi$
$542$			18.3848i			0.789694i
$543$		0			0
$544$			24.0000i			1.02899i
$545$	14.1421			0.605783
$546$		0			0
$547$	−5.00000	+	5.00000i	−0.213785	+	0.213785i	−0.805873	−	0.592088i	$-0.798304\pi$
$547$	−5.00000	+	5.00000i	−0.213785	+	0.213785i	0.592088	+	0.805873i	$0.298304\pi$
$548$		−	36.7696i		−	1.57072i
$549$		0			0
$550$	20.0000	+	20.0000i	0.852803	+	0.852803i
$551$			22.6274i			0.963960i
$552$		0			0
$553$		−	6.00000i		−	0.255146i
$554$	−28.2843	+	28.2843i	−1.20168	+	1.20168i
$555$		0			0
$556$	−20.0000	−	20.0000i	−0.848189	−	0.848189i
$557$	−9.19239	+	9.19239i	−0.389494	+	0.389494i	−0.874507	−	0.485013i	$-0.838815\pi$
$557$	−9.19239	+	9.19239i	−0.389494	+	0.389494i	0.485013	+	0.874507i	$0.338815\pi$
$558$		0			0
$559$	14.0000			0.592137
$560$	8.48528	−	8.48528i	0.358569	−	0.358569i
$561$		0			0
$562$	−34.0000			−1.43420
$563$	−0.707107	+	0.707107i	−0.0298010	+	0.0298010i	−0.721850	−	0.692049i	$-0.756708\pi$
$563$	−0.707107	+	0.707107i	−0.0298010	+	0.0298010i	0.692049	+	0.721850i	$0.256708\pi$
$564$		0			0
$565$	6.00000	+	6.00000i	0.252422	+	0.252422i
$566$	5.65685	−	5.65685i	0.237775	−	0.237775i
$567$		0			0
$568$	44.0000			1.84620
$569$			5.65685i			0.237148i	0.992945	+	0.118574i	$0.0378322\pi$
$569$			5.65685i			0.237148i	−0.992945	+	0.118574i	$0.962168\pi$
$570$		0			0
$571$	7.00000	+	7.00000i	0.292941	+	0.292941i	0.838241	−	0.545300i	$-0.183584\pi$
$571$	7.00000	+	7.00000i	0.292941	+	0.292941i	−0.545300	+	0.838241i	$0.683584\pi$
$572$	14.1421			0.591312
$573$		0			0
$574$		−	24.0000i		−	1.00174i
$575$	−11.3137			−0.471814
$576$		0			0
$577$	32.0000			1.33218			0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000			1.33218			0.666089	+	0.745873i	$0.267967\pi$
$578$		−	1.41421i		−	0.0588235i
$579$		0			0
$580$	8.00000			0.332182
$581$	−23.3345	−	23.3345i	−0.968079	−	0.968079i
$582$		0			0
$583$		−	5.00000i		−	0.207079i
$584$	−42.4264			−1.75562
$585$		0			0
$586$	−20.0000	+	20.0000i	−0.826192	+	0.826192i
$587$	−7.77817	−	7.77817i	−0.321040	−	0.321040i	0.528126	−	0.849166i	$-0.322895\pi$
$587$	−7.77817	−	7.77817i	−0.321040	−	0.321040i	−0.849166	+	0.528126i	$0.822895\pi$
$588$		0			0
$589$	−28.0000	+	28.0000i	−1.15372	+	1.15372i
$590$	14.1421			0.582223
$591$		0			0
$592$	−8.00000	−	8.00000i	−0.328798	−	0.328798i
$593$	−29.6985			−1.21957			−0.609785	−	0.792567i	$-0.708744\pi$
$593$	−29.6985			−1.21957			−0.609785	+	0.792567i	$0.708744\pi$
$594$		0			0
$595$	−9.00000	+	9.00000i	−0.368964	+	0.368964i
$596$	−9.89949	−	9.89949i	−0.405499	−	0.405499i
$597$		0			0
$598$	−4.00000	+	4.00000i	−0.163572	+	0.163572i
$599$		−	41.0122i		−	1.67571i	−0.545891	−	0.837856i	$-0.683809\pi$
$599$		−	41.0122i		−	1.67571i	0.545891	−	0.837856i	$-0.316191\pi$
$600$		0			0
$601$		−	27.0000i		−	1.10135i	−0.834719	−	0.550676i	$-0.814370\pi$
$601$		−	27.0000i		−	1.10135i	0.834719	−	0.550676i	$-0.185630\pi$
$602$	−29.6985	−	29.6985i	−1.21042	−	1.21042i
$603$		0			0
$604$		−	42.0000i		−	1.70896i
$605$	−9.89949	+	9.89949i	−0.402472	+	0.402472i
$606$		0			0
$607$	20.0000			0.811775			0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000			0.811775			0.405887	+	0.913923i	$0.366962\pi$
$608$	−22.6274	−	22.6274i	−0.917663	−	0.917663i
$609$		0			0
$610$		−	20.0000i		−	0.809776i
$611$	4.24264	−	4.24264i	0.171639	−	0.171639i
$612$		0			0
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	0.727013	−	0.686624i	$-0.240908\pi$
$613$	1.00000	+	1.00000i	0.0403896	+	0.0403896i	−0.686624	+	0.727013i	$0.740908\pi$
$614$	−9.89949	−	9.89949i	−0.399511	−	0.399511i
$615$		0			0
$616$	−30.0000	−	30.0000i	−1.20873	−	1.20873i
$617$		−	2.82843i		−	0.113868i	−0.998378	−	0.0569341i	$-0.981868\pi$
$617$		−	2.82843i		−	0.113868i	0.998378	−	0.0569341i	$-0.0181325\pi$
$618$		0			0
$619$	1.00000	+	1.00000i	0.0401934	+	0.0401934i	0.726918	−	0.686724i	$-0.240952\pi$
$619$	1.00000	+	1.00000i	0.0401934	+	0.0401934i	−0.686724	+	0.726918i	$0.740952\pi$
$620$	9.89949	+	9.89949i	0.397573	+	0.397573i
$621$		0			0
$622$	−28.0000			−1.12270
$623$	4.24264			0.169978
$624$		0			0
$625$	−11.0000			−0.440000
$626$	−38.1838			−1.52613
$627$		0			0
$628$	4.00000	−	4.00000i	0.159617	−	0.159617i
$629$	8.48528	+	8.48528i	0.338330	+	0.338330i
$630$		0			0
$631$			15.0000i			0.597141i	0.954388	+	0.298570i	$0.0965097\pi$
$631$			15.0000i			0.597141i	−0.954388	+	0.298570i	$0.903490\pi$
$632$			5.65685i			0.225018i
$633$		0			0
$634$	−35.0000	−	35.0000i	−1.39003	−	1.39003i
$635$	−9.19239	−	9.19239i	−0.364789	−	0.364789i
$636$		0			0
$637$	−2.00000	+	2.00000i	−0.0792429	+	0.0792429i
$638$		−	28.2843i		−	1.11979i
$639$		0			0
$640$	−8.00000	+	8.00000i	−0.316228	+	0.316228i
$641$	42.4264			1.67574			0.837871	−	0.545868i	$-0.183800\pi$
$641$	42.4264			1.67574			0.837871	+	0.545868i	$0.183800\pi$
$642$		0			0
$643$	10.0000	−	10.0000i	0.394362	−	0.394362i	−0.481877	−	0.876239i	$-0.660045\pi$
$643$	10.0000	−	10.0000i	0.394362	−	0.394362i	0.876239	+	0.481877i	$0.160045\pi$
$644$	16.9706			0.668734
$645$		0			0
$646$	24.0000	+	24.0000i	0.944267	+	0.944267i
$647$		−	28.2843i		−	1.11197i	−0.831193	−	0.555985i	$-0.812341\pi$
$647$		−	28.2843i		−	1.11197i	0.831193	−	0.555985i	$-0.187659\pi$
$648$		0			0
$649$		−	50.0000i		−	1.96267i
$650$	−5.65685	+	5.65685i	−0.221880	+	0.221880i
$651$		0			0
$652$	10.0000	−	10.0000i	0.391630	−	0.391630i
$653$	−21.9203	+	21.9203i	−0.857808	+	0.857808i	−0.991080	−	0.133272i	$-0.957452\pi$
$653$	−21.9203	+	21.9203i	−0.857808	+	0.857808i	0.133272	+	0.991080i	$0.457452\pi$
$654$		0			0
$655$	−1.00000			−0.0390732
$656$			22.6274i			0.883452i
$657$		0			0
$658$	−18.0000			−0.701713
$659$	−9.19239	+	9.19239i	−0.358085	+	0.358085i	−0.863107	−	0.505022i	$-0.831484\pi$
$659$	−9.19239	+	9.19239i	−0.358085	+	0.358085i	0.505022	+	0.863107i	$0.331484\pi$
$660$		0			0
$661$	16.0000	+	16.0000i	0.622328	+	0.622328i	0.946126	−	0.323798i	$-0.104960\pi$
$661$	16.0000	+	16.0000i	0.622328	+	0.622328i	−0.323798	+	0.946126i	$0.604960\pi$
$662$	9.89949	−	9.89949i	0.384755	−	0.384755i
$663$		0			0
$664$	22.0000	+	22.0000i	0.853766	+	0.853766i
$665$		−	16.9706i		−	0.658090i
$666$		0			0
$667$	8.00000	+	8.00000i	0.309761	+	0.309761i
$668$			14.1421i			0.547176i
$669$		0			0
$670$		−	2.00000i		−	0.0772667i
$671$	−70.7107			−2.72976
$672$		0			0
$673$	−7.00000			−0.269830			−0.134915	−	0.990857i	$-0.543076\pi$
$673$	−7.00000			−0.269830			−0.134915	+	0.990857i	$0.543076\pi$
$674$		−	28.2843i		−	1.08947i
$675$		0			0
$676$		−	22.0000i		−	0.846154i
$677$	−22.6274	−	22.6274i	−0.869642	−	0.869642i	0.122790	−	0.992433i	$-0.460816\pi$
$677$	−22.6274	−	22.6274i	−0.869642	−	0.869642i	−0.992433	+	0.122790i	$0.960816\pi$
$678$		0			0
$679$			21.0000i			0.805906i
$680$	8.48528	−	8.48528i	0.325396	−	0.325396i
$681$		0			0
$682$	35.0000	−	35.0000i	1.34022	−	1.34022i
$683$	11.3137	+	11.3137i	0.432907	+	0.432907i	0.889616	−	0.456709i	$-0.150972\pi$
$683$	11.3137	+	11.3137i	0.432907	+	0.432907i	−0.456709	+	0.889616i	$0.650972\pi$
$684$		0			0
$685$	−13.0000	+	13.0000i	−0.496704	+	0.496704i
$686$	−21.2132			−0.809924
$687$		0			0
$688$	28.0000	+	28.0000i	1.06749	+	1.06749i
$689$	1.41421			0.0538772
$690$		0			0
$691$	16.0000	−	16.0000i	0.608669	−	0.608669i	−0.333929	−	0.942598i	$-0.608375\pi$
$691$	16.0000	−	16.0000i	0.608669	−	0.608669i	0.942598	+	0.333929i	$0.108375\pi$
$692$	−15.5563	+	15.5563i	−0.591364	+	0.591364i
$693$		0			0
$694$	25.0000	−	25.0000i	0.948987	−	0.948987i
$695$			14.1421i			0.536442i
$696$		0			0
$697$		−	24.0000i		−	0.909065i
$698$	−5.65685	−	5.65685i	−0.214115	−	0.214115i
$699$		0			0
$700$	24.0000			0.907115
$701$	−0.707107	+	0.707107i	−0.0267071	+	0.0267071i	−0.720334	−	0.693627i	$-0.756012\pi$
$701$	−0.707107	+	0.707107i	−0.0267071	+	0.0267071i	0.693627	+	0.720334i	$0.256012\pi$
$702$		0			0
$703$	−16.0000			−0.603451
$704$	28.2843	+	28.2843i	1.06600	+	1.06600i
$705$		0			0
$706$		0			0
$707$	14.8492	−	14.8492i	0.558463	−	0.558463i
$708$		0			0
$709$	22.0000	+	22.0000i	0.826227	+	0.826227i	0.986993	−	0.160765i	$-0.0513962\pi$
$709$	22.0000	+	22.0000i	0.826227	+	0.826227i	−0.160765	+	0.986993i	$0.551396\pi$
$710$	−15.5563	−	15.5563i	−0.583819	−	0.583819i
$711$		0			0
$712$	−4.00000			−0.149906
$713$			19.7990i			0.741478i
$714$		0			0
$715$	−5.00000	−	5.00000i	−0.186989	−	0.186989i
$716$	−15.5563	+	15.5563i	−0.581368	+	0.581368i
$717$		0			0
$718$	50.0000			1.86598
$719$	16.9706			0.632895			0.316448	−	0.948610i	$-0.397510\pi$
$719$	16.9706			0.632895			0.316448	+	0.948610i	$0.397510\pi$
$720$		0			0
$721$	−18.0000			−0.670355
$722$	−18.3848			−0.684211
$723$		0			0
$724$	−32.0000	−	32.0000i	−1.18927	−	1.18927i
$725$	11.3137	+	11.3137i	0.420181	+	0.420181i
$726$		0			0
$727$			9.00000i			0.333792i	0.985975	+	0.166896i	$0.0533743\pi$
$727$			9.00000i			0.333792i	−0.985975	+	0.166896i	$0.946626\pi$
$728$	8.48528	−	8.48528i	0.314485	−	0.314485i
$729$		0			0
$730$	15.0000	+	15.0000i	0.555175	+	0.555175i
$731$	−29.6985	−	29.6985i	−1.09844	−	1.09844i
$732$		0			0
$733$	−26.0000	+	26.0000i	−0.960332	+	0.960332i	−0.999243	−	0.0389108i	$-0.987611\pi$
$733$	−26.0000	+	26.0000i	−0.960332	+	0.960332i	0.0389108	+	0.999243i	$0.487611\pi$
$734$		−	15.5563i		−	0.574195i
$735$		0			0
$736$	−16.0000			−0.589768
$737$	−7.07107			−0.260466
$738$		0			0
$739$	−29.0000	+	29.0000i	−1.06678	+	1.06678i	−0.0691779	+	0.997604i	$0.522038\pi$
$739$	−29.0000	+	29.0000i	−1.06678	+	1.06678i	−0.997604	+	0.0691779i	$0.977962\pi$
$740$			5.65685i			0.207950i
$741$		0			0
$742$	−3.00000	−	3.00000i	−0.110133	−	0.110133i
$743$		−	41.0122i		−	1.50459i	−0.658826	−	0.752296i	$-0.728946\pi$
$743$		−	41.0122i		−	1.50459i	0.658826	−	0.752296i	$-0.271054\pi$
$744$		0			0
$745$			7.00000i			0.256460i
$746$	1.41421	−	1.41421i	0.0517780	−	0.0517780i
$747$		0			0
$748$	−30.0000	−	30.0000i	−1.09691	−	1.09691i
$749$	−23.3345	+	23.3345i	−0.852625	+	0.852625i
$750$		0			0
$751$	−1.00000			−0.0364905			−0.0182453	−	0.999834i	$-0.505808\pi$
$751$	−1.00000			−0.0364905			−0.0182453	+	0.999834i	$0.505808\pi$
$752$	16.9706			0.618853
$753$		0			0
$754$	8.00000			0.291343
$755$	−14.8492	+	14.8492i	−0.540419	+	0.540419i
$756$		0			0
$757$	13.0000	+	13.0000i	0.472493	+	0.472493i	0.902721	−	0.430227i	$-0.141567\pi$
$757$	13.0000	+	13.0000i	0.472493	+	0.472493i	−0.430227	+	0.902721i	$0.641567\pi$
$758$	−2.82843	+	2.82843i	−0.102733	+	0.102733i
$759$		0			0
$760$			16.0000i			0.580381i
$761$			31.1127i			1.12783i	0.825831	+	0.563917i	$0.190706\pi$
$761$			31.1127i			1.12783i	−0.825831	+	0.563917i	$0.809294\pi$
$762$		0			0
$763$	−30.0000	−	30.0000i	−1.08607	−	1.08607i
$764$	−8.48528			−0.306987
$765$		0			0
$766$			6.00000i			0.216789i
$767$	14.1421			0.510643
$768$		0			0
$769$	11.0000			0.396670			0.198335	−	0.980134i	$-0.436447\pi$
$769$	11.0000			0.396670			0.198335	+	0.980134i	$0.436447\pi$
$770$			21.2132i			0.764471i
$771$		0			0
$772$	14.0000			0.503871
$773$	7.07107	+	7.07107i	0.254329	+	0.254329i	0.822743	−	0.568414i	$-0.192443\pi$
$773$	7.07107	+	7.07107i	0.254329	+	0.254329i	−0.568414	+	0.822743i	$0.692443\pi$
$774$		0			0
$775$			28.0000i			1.00579i
$776$		−	19.7990i		−	0.710742i
$777$		0			0
$778$	13.0000	−	13.0000i	0.466073	−	0.466073i
$779$	22.6274	+	22.6274i	0.810711	+	0.810711i
$780$		0			0
$781$	−55.0000	+	55.0000i	−1.96805	+	1.96805i
$782$	16.9706			0.606866
$783$		0			0
$784$	−8.00000			−0.285714
$785$	−2.82843			−0.100951
$786$		0			0
$787$	−26.0000	+	26.0000i	−0.926800	+	0.926800i	−0.997498	−	0.0706979i	$-0.977477\pi$
$787$	−26.0000	+	26.0000i	−0.926800	+	0.926800i	0.0706979	+	0.997498i	$0.477477\pi$
$788$	32.5269	+	32.5269i	1.15872	+	1.15872i
$789$		0			0
$790$	2.00000	−	2.00000i	0.0711568	−	0.0711568i
$791$		−	25.4558i		−	0.905106i
$792$		0			0
$793$		−	20.0000i		−	0.710221i
$794$	7.07107	+	7.07107i	0.250943	+	0.250943i
$795$		0			0
$796$			30.0000i			1.06332i
$797$	16.2635	−	16.2635i	0.576081	−	0.576081i	−0.357740	−	0.933821i	$-0.616453\pi$
$797$	16.2635	−	16.2635i	0.576081	−	0.576081i	0.933821	+	0.357740i	$0.116453\pi$
$798$		0			0
$799$	−18.0000			−0.636794
$800$	−22.6274			−0.800000
$801$		0			0
$802$		−	12.0000i		−	0.423735i
$803$	53.0330	−	53.0330i	1.87149	−	1.87149i
$804$		0			0
$805$	−6.00000	−	6.00000i	−0.211472	−	0.211472i
$806$	9.89949	+	9.89949i	0.348695	+	0.348695i
$807$		0			0
$808$	−14.0000	+	14.0000i	−0.492518	+	0.492518i
$809$			52.3259i			1.83968i	0.392293	+	0.919840i	$0.371682\pi$
$809$			52.3259i			1.83968i	−0.392293	+	0.919840i	$0.628318\pi$
$810$		0			0
$811$	10.0000	+	10.0000i	0.351147	+	0.351147i	0.860536	−	0.509389i	$-0.170129\pi$
$811$	10.0000	+	10.0000i	0.351147	+	0.351147i	−0.509389	+	0.860536i	$0.670129\pi$
$812$	−16.9706	−	16.9706i	−0.595550	−	0.595550i
$813$		0			0
$814$	20.0000			0.701000
$815$	−7.07107			−0.247689
$816$		0			0
$817$	56.0000			1.95919
$818$	−4.24264			−0.148340
$819$		0			0
$820$	8.00000	−	8.00000i	0.279372	−	0.279372i
$821$	32.5269	+	32.5269i	1.13520	+	1.13520i	0.989300	+	0.145897i	$0.0466069\pi$
$821$	32.5269	+	32.5269i	1.13520	+	1.13520i	0.145897	+	0.989300i	$0.453393\pi$
$822$		0			0
$823$			9.00000i			0.313720i	0.987621	+	0.156860i	$0.0501372\pi$
$823$			9.00000i			0.313720i	−0.987621	+	0.156860i	$0.949863\pi$
$824$	16.9706			0.591198
$825$		0			0
$826$	−30.0000	−	30.0000i	−1.04383	−	1.04383i
$827$	15.5563	+	15.5563i	0.540947	+	0.540947i	0.923807	−	0.382859i	$-0.125061\pi$
$827$	15.5563	+	15.5563i	0.540947	+	0.540947i	−0.382859	+	0.923807i	$0.625061\pi$
$828$		0			0
$829$	−11.0000	+	11.0000i	−0.382046	+	0.382046i	−0.871839	−	0.489793i	$-0.837072\pi$
$829$	−11.0000	+	11.0000i	−0.382046	+	0.382046i	0.489793	+	0.871839i	$0.337072\pi$
$830$		−	15.5563i		−	0.539969i
$831$		0			0
$832$	−8.00000	+	8.00000i	−0.277350	+	0.277350i
$833$	8.48528			0.293998
$834$		0			0
$835$	5.00000	−	5.00000i	0.173032	−	0.173032i
$836$	56.5685			1.95646
$837$		0			0
$838$	10.0000	+	10.0000i	0.345444	+	0.345444i
$839$			9.89949i			0.341769i	0.985291	+	0.170884i	$0.0546624\pi$
$839$			9.89949i			0.341769i	−0.985291	+	0.170884i	$0.945338\pi$
$840$		0			0
$841$			13.0000i			0.448276i
$842$	22.6274	−	22.6274i	0.779792	−	0.779792i
$843$		0			0
$844$	−8.00000	+	8.00000i	−0.275371	+	0.275371i
$845$	−7.77817	+	7.77817i	−0.267577	+	0.267577i
$846$		0			0
$847$	42.0000			1.44314
$848$	2.82843	+	2.82843i	0.0971286	+	0.0971286i
$849$		0			0
$850$	24.0000			0.823193
$851$	−5.65685	+	5.65685i	−0.193914	+	0.193914i
$852$		0			0
$853$	−5.00000	−	5.00000i	−0.171197	−	0.171197i	0.616308	−	0.787505i	$-0.288628\pi$
$853$	−5.00000	−	5.00000i	−0.171197	−	0.171197i	−0.787505	+	0.616308i	$0.788628\pi$
$854$	−42.4264	+	42.4264i	−1.45180	+	1.45180i
$855$		0			0
$856$	22.0000	−	22.0000i	0.751945	−	0.751945i
$857$		−	32.5269i		−	1.11110i	−0.831484	−	0.555549i	$-0.812508\pi$
$857$		−	32.5269i		−	1.11110i	0.831484	−	0.555549i	$-0.187492\pi$
$858$		0			0
$859$	4.00000	+	4.00000i	0.136478	+	0.136478i	0.772046	−	0.635567i	$-0.219234\pi$
$859$	4.00000	+	4.00000i	0.136478	+	0.136478i	−0.635567	+	0.772046i	$0.719234\pi$
$860$		−	19.7990i		−	0.675140i
$861$		0			0
$862$			6.00000i			0.204361i
$863$	16.9706			0.577685			0.288842	−	0.957377i	$-0.406730\pi$
$863$	16.9706			0.577685			0.288842	+	0.957377i	$0.406730\pi$
$864$		0			0
$865$	11.0000			0.374011
$866$		−	7.07107i		−	0.240285i
$867$		0			0
$868$		−	42.0000i		−	1.42557i
$869$	−7.07107	−	7.07107i	−0.239870	−	0.239870i
$870$		0			0
$871$		−	2.00000i		−	0.0677674i
$872$	28.2843	+	28.2843i	0.957826	+	0.957826i
$873$		0			0
$874$	−16.0000	+	16.0000i	−0.541208	+	0.541208i
$875$	−19.0919	−	19.0919i	−0.645423	−	0.645423i
$876$		0			0
$877$	−11.0000	+	11.0000i	−0.371444	+	0.371444i	−0.868003	−	0.496559i	$-0.834597\pi$
$877$	−11.0000	+	11.0000i	−0.371444	+	0.371444i	0.496559	+	0.868003i	$0.334597\pi$
$878$	−21.2132			−0.715911
$879$		0			0
$880$		−	20.0000i		−	0.674200i
$881$	−12.7279			−0.428815			−0.214407	−	0.976744i	$-0.568782\pi$
$881$	−12.7279			−0.428815			−0.214407	+	0.976744i	$0.568782\pi$
$882$		0			0
$883$	−26.0000	+	26.0000i	−0.874970	+	0.874970i	−0.993009	−	0.118039i	$-0.962339\pi$
$883$	−26.0000	+	26.0000i	−0.874970	+	0.874970i	0.118039	+	0.993009i	$0.462339\pi$
$884$	8.48528	−	8.48528i	0.285391	−	0.285391i
$885$		0			0
$886$	−14.0000	+	14.0000i	−0.470339	+	0.470339i
$887$			9.89949i			0.332393i	0.986093	+	0.166196i	$0.0531485\pi$
$887$			9.89949i			0.332393i	−0.986093	+	0.166196i	$0.946851\pi$
$888$		0			0
$889$			39.0000i			1.30802i
$890$	1.41421	+	1.41421i	0.0474045	+	0.0474045i
$891$		0			0
$892$	44.0000			1.47323
$893$	16.9706	−	16.9706i	0.567898	−	0.567898i
$894$		0			0
$895$	11.0000			0.367689
$896$	33.9411			1.13389
$897$		0			0
$898$		−	24.0000i		−	0.800890i
$899$	19.7990	−	19.7990i	0.660333	−	0.660333i
$900$		0			0
$901$	−3.00000	−	3.00000i	−0.0999445	−	0.0999445i
$902$	−28.2843	−	28.2843i	−0.941763	−	0.941763i
$903$		0			0
$904$			24.0000i			0.798228i
$905$			22.6274i			0.752161i
$906$		0			0
$907$	−8.00000	−	8.00000i	−0.265636	−	0.265636i	0.561703	−	0.827339i	$-0.310146\pi$
$907$	−8.00000	−	8.00000i	−0.265636	−	0.265636i	−0.827339	+	0.561703i	$0.810146\pi$
$908$	14.1421	−	14.1421i	0.469323	−	0.469323i
$909$		0			0
$910$	−6.00000			−0.198898
$911$	38.1838			1.26508			0.632542	−	0.774526i	$-0.282011\pi$
$911$	38.1838			1.26508			0.632542	+	0.774526i	$0.282011\pi$
$912$		0			0
$913$	−55.0000			−1.82023
$914$	4.24264			0.140334
$915$		0			0
$916$	−14.0000	−	14.0000i	−0.462573	−	0.462573i
$917$	2.12132	+	2.12132i	0.0700522	+	0.0700522i
$918$		0			0
$919$			15.0000i			0.494804i	0.968913	+	0.247402i	$0.0795769\pi$
$919$			15.0000i			0.494804i	−0.968913	+	0.247402i	$0.920423\pi$
$920$	5.65685	+	5.65685i	0.186501	+	0.186501i
$921$		0			0
$922$	−35.0000	−	35.0000i	−1.15266	−	1.15266i
$923$	−15.5563	−	15.5563i	−0.512043	−	0.512043i
$924$		0			0
$925$	−8.00000	+	8.00000i	−0.263038	+	0.263038i
$926$			9.89949i			0.325318i
$927$		0			0
$928$	16.0000	+	16.0000i	0.525226	+	0.525226i
$929$	−42.4264			−1.39197			−0.695983	−	0.718059i	$-0.745031\pi$
$929$	−42.4264			−1.39197			−0.695983	+	0.718059i	$0.745031\pi$
$930$		0			0
$931$	−8.00000	+	8.00000i	−0.262189	+	0.262189i
$932$			31.1127i			1.01913i
$933$		0			0
$934$	−5.00000	−	5.00000i	−0.163605	−	0.163605i
$935$			21.2132i			0.693746i
$936$		0			0
$937$			3.00000i			0.0980057i	0.998799	+	0.0490029i	$0.0156043\pi$
$937$			3.00000i			0.0980057i	−0.998799	+	0.0490029i	$0.984396\pi$
$938$	−4.24264	+	4.24264i	−0.138527	+	0.138527i
$939$		0			0
$940$	−6.00000	−	6.00000i	−0.195698	−	0.195698i
$941$	−21.9203	+	21.9203i	−0.714582	+	0.714582i	−0.967490	−	0.252909i	$-0.918613\pi$
$941$	−21.9203	+	21.9203i	−0.714582	+	0.714582i	0.252909	+	0.967490i	$0.418613\pi$
$942$		0			0
$943$	16.0000			0.521032
$944$	28.2843	+	28.2843i	0.920575	+	0.920575i
$945$		0			0
$946$	−70.0000			−2.27590
$947$	−9.19239	+	9.19239i	−0.298712	+	0.298712i	−0.840509	−	0.541797i	$-0.817744\pi$
$947$	−9.19239	+	9.19239i	−0.298712	+	0.298712i	0.541797	+	0.840509i	$0.317744\pi$
$948$		0			0
$949$	15.0000	+	15.0000i	0.486921	+	0.486921i
$950$	−22.6274	+	22.6274i	−0.734130	+	0.734130i
$951$		0			0
$952$	−36.0000			−1.16677
$953$			5.65685i			0.183243i	0.995794	+	0.0916217i	$0.0292051\pi$
$953$			5.65685i			0.183243i	−0.995794	+	0.0916217i	$0.970795\pi$
$954$		0			0
$955$	3.00000	+	3.00000i	0.0970777	+	0.0970777i
$956$	50.9117			1.64660
$957$		0			0
$958$			6.00000i			0.193851i
$959$	55.1543			1.78103
$960$		0			0
$961$	18.0000			0.580645
$962$			5.65685i			0.182384i
$963$		0			0
$964$	8.00000			0.257663
$965$	−4.94975	−	4.94975i	−0.159338	−	0.159338i
$966$		0			0
$967$			33.0000i			1.06121i	0.847620	+	0.530604i	$0.178035\pi$
$967$			33.0000i			1.06121i	−0.847620	+	0.530604i	$0.821965\pi$
$968$	−39.5980			−1.27273
$969$		0			0
$970$	−7.00000	+	7.00000i	−0.224756	+	0.224756i
$971$	−24.7487	−	24.7487i	−0.794225	−	0.794225i	0.187953	−	0.982178i	$-0.439815\pi$
$971$	−24.7487	−	24.7487i	−0.794225	−	0.794225i	−0.982178	+	0.187953i	$0.939815\pi$
$972$		0			0
$973$	30.0000	−	30.0000i	0.961756	−	0.961756i
$974$	16.9706			0.543772
$975$		0			0
$976$	40.0000	−	40.0000i	1.28037	−	1.28037i
$977$	−38.1838			−1.22161			−0.610803	−	0.791782i	$-0.709153\pi$
$977$	−38.1838			−1.22161			−0.610803	+	0.791782i	$0.709153\pi$
$978$		0			0
$979$	5.00000	−	5.00000i	0.159801	−	0.159801i
$980$	2.82843	+	2.82843i	0.0903508	+	0.0903508i
$981$		0			0
$982$	7.00000	−	7.00000i	0.223379	−	0.223379i
$983$		−	32.5269i		−	1.03745i	−0.854942	−	0.518724i	$-0.826407\pi$
$983$		−	32.5269i		−	1.03745i	0.854942	−	0.518724i	$-0.173593\pi$
$984$		0			0
$985$		−	23.0000i		−	0.732841i
$986$	−16.9706	−	16.9706i	−0.540453	−	0.540453i
$987$		0			0
$988$			16.0000i			0.509028i
$989$	19.7990	−	19.7990i	0.629571	−	0.629571i
$990$		0			0
$991$	53.0000			1.68360			0.841800	−	0.539789i	$-0.181496\pi$
$991$	53.0000			1.68360			0.841800	+	0.539789i	$0.181496\pi$
$992$			39.5980i			1.25724i
$993$		0			0
$994$			66.0000i			2.09339i
$995$	10.6066	−	10.6066i	0.336252	−	0.336252i
$996$		0			0
$997$	7.00000	+	7.00000i	0.221692	+	0.221692i	0.809211	−	0.587519i	$-0.199895\pi$
$997$	7.00000	+	7.00000i	0.221692	+	0.221692i	−0.587519	+	0.809211i	$0.699895\pi$
$998$	32.5269	+	32.5269i	1.02962	+	1.02962i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.k.b.325.1	yes	4
3.2	odd	2	inner	432.2.k.b.325.2	yes	4
4.3	odd	2		1728.2.k.b.433.2		4
12.11	even	2		1728.2.k.b.433.1		4
16.3	odd	4		1728.2.k.b.1297.2		4
16.13	even	4	inner	432.2.k.b.109.2	yes	4
48.29	odd	4	inner	432.2.k.b.109.1	✓	4
48.35	even	4		1728.2.k.b.1297.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.k.b.109.1	✓	4	48.29	odd	4	inner
432.2.k.b.109.2	yes	4	16.13	even	4	inner
432.2.k.b.325.1	yes	4	1.1	even	1	trivial
432.2.k.b.325.2	yes	4	3.2	odd	2	inner
1728.2.k.b.433.1		4	12.11	even	2
1728.2.k.b.433.2		4	4.3	odd	2
1728.2.k.b.1297.1		4	48.35	even	4
1728.2.k.b.1297.2		4	16.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +(0.707107 + 0.707107i) q^{5} -3.00000i q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +(0.707107 + 0.707107i) q^{5} -3.00000i q^{7} +2.82843i q^{8} +(1.00000 - 1.00000i) q^{10} +(-3.53553 - 3.53553i) q^{11} +(1.00000 - 1.00000i) q^{13} -4.24264 q^{14} +4.00000 q^{16} -4.24264 q^{17} +(4.00000 - 4.00000i) q^{19} +(-1.41421 - 1.41421i) q^{20} +(-5.00000 + 5.00000i) q^{22} -2.82843i q^{23} -4.00000i q^{25} +(-1.41421 - 1.41421i) q^{26} +6.00000i q^{28} +(-2.82843 + 2.82843i) q^{29} -7.00000 q^{31} -5.65685i q^{32} +6.00000i q^{34} +(2.12132 - 2.12132i) q^{35} +(-2.00000 - 2.00000i) q^{37} +(-5.65685 - 5.65685i) q^{38} +(-2.00000 + 2.00000i) q^{40} +5.65685i q^{41} +(7.00000 + 7.00000i) q^{43} +(7.07107 + 7.07107i) q^{44} -4.00000 q^{46} +4.24264 q^{47} -2.00000 q^{49} -5.65685 q^{50} +(-2.00000 + 2.00000i) q^{52} +(0.707107 + 0.707107i) q^{53} -5.00000i q^{55} +8.48528 q^{56} +(4.00000 + 4.00000i) q^{58} +(7.07107 + 7.07107i) q^{59} +(10.0000 - 10.0000i) q^{61} +9.89949i q^{62} -8.00000 q^{64} +1.41421 q^{65} +(1.00000 - 1.00000i) q^{67} +8.48528 q^{68} +(-3.00000 - 3.00000i) q^{70} -15.5563i q^{71} +15.0000i q^{73} +(-2.82843 + 2.82843i) q^{74} +(-8.00000 + 8.00000i) q^{76} +(-10.6066 + 10.6066i) q^{77} +2.00000 q^{79} +(2.82843 + 2.82843i) q^{80} +8.00000 q^{82} +(7.77817 - 7.77817i) q^{83} +(-3.00000 - 3.00000i) q^{85} +(9.89949 - 9.89949i) q^{86} +(10.0000 - 10.0000i) q^{88} +1.41421i q^{89} +(-3.00000 - 3.00000i) q^{91} +5.65685i q^{92} -6.00000i q^{94} +5.65685 q^{95} -7.00000 q^{97} +2.82843i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} + 4 q^{10} + 4 q^{13} + 16 q^{16} + 16 q^{19} - 20 q^{22} - 28 q^{31} - 8 q^{37} - 8 q^{40} + 28 q^{43} - 16 q^{46} - 8 q^{49} - 8 q^{52} + 16 q^{58} + 40 q^{61} - 32 q^{64} + 4 q^{67} - 12 q^{70} - 32 q^{76} + 8 q^{79} + 32 q^{82} - 12 q^{85} + 40 q^{88} - 12 q^{91} - 28 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 4 * q^10 + 4 * q^13 + 16 * q^16 + 16 * q^19 - 20 * q^22 - 28 * q^31 - 8 * q^37 - 8 * q^40 + 28 * q^43 - 16 * q^46 - 8 * q^49 - 8 * q^52 + 16 * q^58 + 40 * q^61 - 32 * q^64 + 4 * q^67 - 12 * q^70 - 32 * q^76 + 8 * q^79 + 32 * q^82 - 12 * q^85 + 40 * q^88 - 12 * q^91 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more